reserve k, m, n, p, K, N for Nat;
reserve i for Integer;
reserve x, y, eps for Real;
reserve seq, seq1, seq2 for Real_Sequence;
reserve sq for FinSequence of REAL;

theorem Th9:
  for seq st for n being Nat holds seq.n=x holds seq is convergent & lim(seq)=x
proof
  let seq;
  assume
A1: for n being Nat holds seq.n=x;
  x in REAL by XREAL_0:def 1;
  then
A2: seq is constant by A1,VALUED_0:def 18;
  hence seq is convergent;
  thus lim(seq) = seq.0 by A2,SEQ_4:26
    .= x by A1;
end;
